In [1]}, Bryant has also given an integral formula for \(\operatorname{\textsl {G}}_2\) -structures. This is expressed in terms of \(\tau _0\) , \(\tau _1\) , \(\tau _2\)  and \(\tau _3\)  which are defined by
\(d\varphi &=\tau _0\, \underline{\star }\, \varphi +3\tau _1\wedge \varphi +\underline{\star }\,\tau _3, \\d\,\underline{\star }\, \varphi &=4\tau _1\wedge \underline{\star }\,\varphi +\tau _2\wedge \varphi .\) 
